Thermal rectification in asymmetric two-phase nanowires

Thermal rectification based on a new strategy of nanostructuration has been assessed with nonequilibrium molecular dynamics and wave-packet propagation simulations. The use of asymmetric crystalline-core/amorphous-conical-shell nanowires creates a direction-dependent thermal conductivity due to variable axial and radial phonon propagation/confinement. The origin of this effect is related to the combination of a crystalline/amorphous interface parallel to the heat flux, together with the variable amount of amorphous coating due to the conical shell of the nanowire. Physical insights of the rectification are given by the mean free path of phonons, the axial and radial energy transfer, the energy diffusivity, and the vibrational density of states restricted to different constitutive elements of the nanowire.

Figure 1

Cross-sectional view along the growth direction of the CL-CS-NW (a), the CO-CS-NWs [(b)–(e)] with four opening angles of the amorphous shell, and, finally, the TE-CS-NW (f). Images are obtained from OVITO MD visualization software [56]. The red colored atoms correspond to the crystalline phase, while the blue ones to the amorphous phase.

Figure 2

Schematic representation of the NWs, with the thermostats ($T_1, T_2$) and fixed atomic positions (f) on the conical CS configurations. Several characteristic elements, such as the effective, core, and shell radii, and the angle of the amorphous shell opening and large (L) and small (S) extremities of the NW are noted along with the flux direction $J_{SL}$ and $J_{LS}$.

Figure 3

The envelope is defined as the maximal amplitude of the wave at a given position (black solid line), and the energy distribution for some time steps represented with different colors. The dashed line corresponds to the fit used to determine the MFP. Here the reference (0) position is the position where the excitation is created, and the initial time corresponds to the Eq. (3).

Figure 4

Power exchanged and rectification for all the 50-nm-long NWs as a function (a) of the opening angle of the amorphous shell (b) of amorphous fraction or thermal conductivity and rectification as a function (c) of the opening angle computed with the effective section defined in Table 1 or (d) with the section next to the hot thermostat. The different configuration are identified by their opening angles for the CO-CS-NW and by TE or CL for the TE-CS-NW and the CL-CS-NW. On the left $y$ axis, ${\kappa }_{LS}$ or $J_{LS}$ are represented by blue stars and ${\kappa }_{SL}$ or $J_{SL}$ by red circles. The rectification values on the right $y$ axis are represented by green triangles, with error bars determined from Eq. (2), and the green dashed line gives the zero rectification.

Figure 5

Longitudinal (first row) and transverse (second row) MFP, diffusivity (third row), and VDOS (last row) for all studied configurations. The solid colored lines represent the SL direction and the dashed ones the LS direction. The different configuration are identified by their opening angles for the CO-CS-NW and by CL for the CL-CS-NW.

Figure 6

Evolution of the mean kinetic energy per atom as function of the radius in 7-nm-thick slice, 9 nm away from the excitation source after a longitudinal impulsion at different times, for the CO-CS-NW-8.8 (upper row) and the CL-CS-NW (lower row). The energy values are normalized by the maximum value for at each given frequency. The black dotted lines represent the density of atoms in the slice and the vertical blue line represents the amorphous/crystalline interface.

Figure 7

Vibrational density of state as a function of the atoms belonging to a cylindrical ring of the crystalline part (four columns) and for different position along the CO-CS-NW-4.4 (colored lines). The insets link the column and the considered radius ranges. The black circle in the inset represent the part of the nanowire core considered, the gray the whole core, and the outer shell is represented by the purple outer circle and is not to scale. The mixed line represents the crystalline amorphous interface.

Figure 8

Vibrational density of state as a function of coaxial (hollow) cylinders of thickness of 0.5 nm for the CL-CS-NW, compared with the bulk VDOS of a-Si and c-Si in dashed lines. The inset links the colors and the considered cylinders, the dot-dashed line represent the crystalline amorphous interface.

Figure 9

Kinetic energy ratio: Sum of atomic kinetic energy in the SL direction ($e_{SL}$, right of the impulsion) divided by the sum of atomic kinetic energy in the LS direction ($e_{LS}$, left of the impulsion), and the mean from the end of the impulsion of propagation until the wave front reaches the boundaries. For the longitudinal (left column) and transverse (right column) excitation.

Figure 10

Evolution of the kinetic energy distribution along the wire (sum of atomic kinetic energy in each individual slices filtered with a polynomial filter) after a longitudinal (upper figure) or transverse (lower figure) impulsion for the CO-CS-NW-8.8. The total energy is separated in the core and the shell at 2 $\text{THz}$ (first row of each subfigure) and 12 $\text{THz}$ for the longitudinal impulsion (second row of the upper figure) and 4 $\text{THz}$ for the transverse one (second row of the lower figure). The energy is normalized by the maximum value for the whole NW.

Figure 11

Energy exchanged by the thermostats, for the CL-CS-NW (a) and the CO-CS-NW-8.8 (b): The first row represents the energy exchanged at each thermostat for five runs, the blue lines represent the energy retrieved by the cold thermostats, and the red lines the energy injected by the hot thermostats. The dashed lines distinguish the SL configuration from the LS configuration represented by the continuous line. The second rows represent the estimation of the energy flux, computed 1000 to 2000 $\mathrm{ps}$, the red symbols represent the hot thermostats, the blue symbols represent the cold thermostats. In the third row the black symbols represent flux taken as the average between hot and cold thermostat. The inverted triangles represent the LS case and the crosses the SL.

Figure 12

$S(\mathbf{q},\omega )$ for (a) longitudinal and (b) transverse polarization in the $\mathrm{\Gamma }X$ direction for the CO-CS-NW-4.4 computed with Eq. (B3) and convoluted with a typical energy resolution curve of linewidth 0.33 $\text{THz}$ to smooth the obtained distribution [42].

Figure 13

Thermal conductivity of propagative modes ${\kappa }_{\mathrm{prop}}$ (in black), and thermal conductivity of the diffusive modes ${\kappa }_{\mathrm{diff}}$ (in red) and total thermal conductivity ${\kappa }_{\mathrm{prop}}+{\kappa }_{\mathrm{diff}}$ (in blue) for the CS-NW-CO-4.4, as a function of temperature in the left panel and contribution of the different frequencies at 300 K in the right panel. The dashed and plain lines represents respectively the LS and SL directions.